Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2961 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#424 | C2 |
| prime radical | C23×C3 | C22×C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #424 | D3 |
| hypocenter | C3 | #2961 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #424 | |
348 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(13,18)(17,23)(19,25)(24,27);(2,3)(12,14,15,21,16,20,22,26)(13,17,19,24,18,23,25,27);(2,3)(4,6,7,11)(5,8,9,10)(14,20)(17,23)(21,26)(24,27);(4,5,7,9)(6,10,11,8)(12,13,15,19,16,18,22,25)(14,17,21,24,20,23,26,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,c^2=ef,f^2=g,b^a=be,d^a=dg,c^b=cg,h^b=h^2,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10704698, #10803036, #10803038, … |
| maximal normal subgroup of | #10704698, #10803036, #10803038, … |
| maximal quotient of | #10793146, #10793150, #10793159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11653);